Problem: We know that $ \frac{n}{n^2-\cos^2(n)}>\frac{n}{n^2}=\frac{1}{n}>0$ for any $n\ge 1$. Considering this fact, what does the direct comparison test say about $\sum\limits_{n=1}^{\infty }~{\frac{n}{n^2-\cos^2n}}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A The series converges. (Choice B) B The series diverges. (Choice C) C The test is inconclusive.
Answer: $\sum\limits_{n=1}^{\infty }~{\frac{1}{{n}}}~$ is the harmonic series which is known to diverge. Our given series is term-by-term greater than a divergent series, so it also diverges.